The first statement is Lemma 1 in [1]}. The second follows by the Cauchy-Schwarz lower bound on \(E_+(A,\xi A)\)  stated in (REF ):
\(\sum _{\xi \in S} \left(E_+(A, \xi A) - {p}\right)& \overset{ (\ref {CS})}{\le }\sum _{0\ne \xi \in \mathbb {F}_p} \left(E_+(A, \xi A) - {p}\right) \\& = \sum _{0\ne \xi \in \mathbb {F}_p} E_+(A, \xi A) - {p} |A|^4 \\& = |A|^4 + p |A|^2 - 2 |A|^3 - |A|^4 + {p} \\& \le p|A|^2.\) 
